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## Numbers and Algebra

### Fundamental Algebra - Algebraic Operations (Arithmetic, Exponents, Roots)

#### Q.02

'Please calculate the multiplication of the following complex numbers:\n(a + bi) * (c + di)\nwhere a, b, c, and d are real numbers.'

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'Find the general term of the sequences {a_{n}}, {b_{n}} defined by the following conditions.'

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'Calculate the expression of sequence a_n according to the following rules: initial condition a_1=5, if even then a_n/2, if odd then a_n+1.'

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'Please calculate the following expression: (6) [\x0crac{x+11}{2 x^{2}+7 x+3}-\x0crac{x-10}{2 x^{2}-3 x-2}] = ?'

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'Describe the properties of an identity equation.'

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'Using the relationship between roots and coefficients, find the following values.'

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'Find the general term of the sequences {a_{n}}, {b_{n}} determined by the following conditions.'

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'(8) \\frac{\\sqrt[3]{a^{4}}}{\\sqrt{b}} \\times \\frac{\\sqrt[3]{b}}{\\sqrt[3]{a^{2}}} \\times \\sqrt[3]{a \\sqrt{b}}=a^{\\frac{4}{3}} b^{-\\frac{1}{2}} \\times a^{-\\frac{2}{3}} b^{\\frac{1}{3}} \\times a^{\\frac{1}{3}} b^{\\frac{1}{6}}'

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'Please calculate the following expression: (4) [(x+y)/(x-y)-(y)/(x-y)+(2x-y)/(y-x)] = ?'

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'(7)\n\\[\egin{aligned}\n\\sqrt[3]{54}+\\sqrt[3]{-250}-\\sqrt[3]{-16} & =\\sqrt[3]{54}-\\sqrt[3]{250}-(-\\sqrt[3]{16}) \n& =\\sqrt[3]{3^{3} \\cdot 2}-\\sqrt[3]{5^{3} \\cdot 2}+\\sqrt[3]{2^{3} \\cdot 2} \n& =3 \\sqrt[3]{2}-5 \\sqrt[3]{2}+2 \\sqrt[3]{2}=(3-5+2) \\sqrt[3]{2} \n& =0\n\\end{aligned}\\]'

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'Practice: If the remainder of dividing a polynomial P(x) by (x+1)^2 is 18x+9, and the remainder of dividing by x-2 is 30 with a remainder of 9, find the remainder when P(x) is divided by (x+1)^2(x-2).'

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'Find the general term of the sequence {a_{n}}, {b_{n}} determined by the following conditions.'

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'Calculate the following expressions. (1) $\\frac{x^{2}+2x+3}{x}-\\frac{x^{2}+3x+5}{x+1}$ (2) $\\frac{x+2}{x}-\\frac{x+3}{x+1}-\\frac{x-5}{x-3}+\\frac{x-6}{x-4}$'

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'Translate the given text into multiple languages.'

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'Perform the following calculation using complex number arithmetic: (3 + 4i) + (1 - 2i).'

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'Practice (1) Let n be a natural number. Find the remainder when x^n-3^n is divided by (x-3)^2. Also, find the remainder when 31x^n-3^n is divided by x^2-5x+6. In (2), find the remainder when 3x^100+2x^97+1 is divided by x^2+1.'

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'Given that a = -1 + √3, find the minimum value of -4√3 + 11.'

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'(1) \ \\frac{8 x^{3} z}{9 b c^{3}} \\times \\frac{27 a b c}{4 x y z^{2}}=\\frac{6 a x^{2}}{c^{2} \oldsymbol{y z}} \\n(2) \\( \\frac{4 a^{2}-b^{2}}{a^{2}-4 b^{2}} \\div \\frac{2 a+b}{a-2 b}=\\frac{(2 a+b)(2 a-b)}{(a+2 b)(a-2 b)} \\times \\frac{a-2 b}{2 a+b} \\)\n\=\\frac{2 a-b}{a+2 b}\\n(3) \\( \\frac{x^{2}}{x+1}-\\frac{1}{x+1}=\\frac{x^{2}-1}{x+1}=\\frac{(x+1)(x-1)}{x+1}=x-1 \\)'

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'Find the sum and product of the given complex number and its conjugate for each.'

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'Find the general term of the sequences \ \\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\} \ defined by the following conditions.\\n\\n\\\na_{1}=1, \\quad b_{1}=-1, \\quad a_{n+1}=5 a_{n}-4 b_{n}, b_{n+1}=a_{n}+b_{n}\\n\'

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'Show the quotient and remainder of polynomial division.'

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'The sequence {p a_{n}+q b_{n}} is an arithmetic sequence, with the first term being p a_{1}+q b_{1}=p a+q b, and the common difference being p d+q e'

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'Find the general term of the sequence \ \\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\} \ defined by the following conditions.'

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'Find the 5th term of the sequence {a_n} defined by the following conditions:'

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'Find the sum of the first 30 terms of an arithmetic sequence with initial term 100 and common difference -8.'

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'Find the general term of the sequence $b_{n+1}=-a_{n}+b_{n}$.'

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'Find the sum S of the 15th to 30th terms of an arithmetic sequence with a common difference of 1, where the 10th term is 1 and the 16th term is 5.'

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'Find the remainder of the given equation by substituting the polynomial at x = i.'

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'Using long division, find the quotient and remainder when the following equations are divided by the 1st degree equation inside [ ].'

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'Find the quotient Q(x) and remainder ax + b when dividing x^{2020} + x^{2021} by x^2 + x + 1'

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'Practice 29: Solve the following problem. Given a polynomial $P(x)$, when divided by $(x-1)(2x+1)$, the quotient is $Q(x)$ and the remainder is $ax+b$. The following equation holds: $P(x)=(x-1)(2x+1)Q(x)+ax+b$.'

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'(2) \\n\\\\[\\\egin{aligned} \\n a^{2} \\times\\left(a^{-1}\\right)^{3} \\div a^{-2} & =a^{2} \\times a^{(-1) \\times 3} \\div a^{-2} \\n & =a^{2-3-(-2)}=a \\n\\\\end{aligned}\\\\]'

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'Adding c to both sides of the equation a>b yields a+c>b+c, adding b to both sides of the equation c>d yields b+c>b+d, hence a+c > b+d'

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'Please calculate the addition of the following complex numbers:\n(a + bi) + (c + di)\nwhere a, b, c, and d are real numbers.'

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'Translate the given text into multiple languages.'

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'Please calculate the subtraction of the following complex numbers.\n(a + bi) - (c + di)\nwhere a, b, c, d are real numbers.'

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'Find the remainder when the following polynomials are divided by the linear expression in [ ].'

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'Express the relative sizes of the following sets of numbers using inequality symbols. \ 1.5, \\log _{2} 5, \\log _{4} 9 \'

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'Practice book 104 p.208\n(1) \ t=\\sin \\theta-\\cos \\theta \ Squaring both sides gives \ t^{2}=\\sin ^{2} \\theta-2 \\sin \\theta \\cos \\theta+\\cos ^{2} \\theta \ Hence, \ t^{2}=1-\\sin 2 \\theta \ and therefore \ \\sin 2 \\theta=-t^{2}+1 \ Thus, \\( f(\\theta)=\\frac{1}{\\sqrt{2}}\\left(-t^{2}+1\\right)-t=-\\frac{\\sqrt{2}}{2} t^{2}-t+\\frac{\\sqrt{2}}{2} \\)'

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'Find the sum of 22: 4 * 1 + 8 * 3 + 12 * 3^2 + ... + 4n * 3^(n-1)'

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'Find the general term of the sequence {an} determined by the following conditions.'

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'Express the order of each set of numbers using inequality symbols.'

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"In example 108, A solved the problem by setting x+y=k. It is impossible to calculate the maximum and minimum values of x+y for all (x, y) pairs contained in region D. Therefore... by setting x+y=k and treating (x, y) as points on the line y=-x+k. Consequently, when the line y=-x+k passes through points in region D (= when it shares points with region D), it is sufficient to consider the maximum and minimum values of the y-intercept k. Now, under the same conditions, let's consider the maximum and minimum values of 2x+y. Setting 2x+y=k and examining the movement of the line y=-2x+k, it is found that the minimum value of k is, similarly to A, when the line (2) passes through the origin O. However, the maximum value of k is not when the line (2) passes through the point (2,2), but when it passes through the point (10/3, 0)."

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'Find the general term of the sequence \ \\left\\{a_{n}\\right\\} \ given that the sum of the terms from the first one to the \ n \-th one is represented by the following formula:'

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'Using available proof methods, prove that the following equation holds true.'

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'Explain the method for division of complex numbers.'

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'Given 106^4 * \\log_{10} 2 = 0.3010, and \\log_{10} 3 = 0.4771. How many digits is 2011? Also, find the leading digit of 2^{2011}.'

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Join our Discord#### Q.67

'Find the remainder when the polynomial P(x)=2 x^{3}-3 x+1 is divided by the following linear expressions: (a) x-1 (b) 2 x+1'

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'Exponents of rational numbers\nFor \ a>0, m, n \ as positive integers, and \ r \ as a positive rational number, then \\( a^{\\frac{1}{n}}=\\sqrt[n]{a}, \\quad a^{\\frac{m}{n}}=(\\sqrt[n]{a})^{m}=\\sqrt[n]{a^{m}}, \\quad a^{-r}=\\frac{1}{a^{r}} \\)'

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'Find the quotient Q and remainder R when the polynomial A is divided by B. Also, express the result in the form of (2) A = BQ + R. (1) A = 4x^3 - 3x - 9, B = 2x + 3 (2) A = 1 + 2x^2 + 2x^3, B = 1 + 2x'

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'Given that the remainder is 17 when the polynomial P(x)=\\frac{1}{2}x^{3}+ax+a^{2}-20 is divided by x-4, find the value of the constant a.'

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"The question's straight lines can be drawn as 2 lines shown in the figure, so the angle between the straight line and the positive direction of the x-axis is \ \\frac{\\pi}{6}+\\frac{\\pi}{4} \\text { or } \\frac{\\pi}{6}-\\frac{\\pi}{4} \"

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'Given text to be translated into multiple languages.'

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'Prove that the equation a^{3}+b^{3}+c^{3}-3abc=0 holds true when a+b+c=0.'

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'Find the remainder when the polynomial $P(x)=x^{3}-2 x^{2}+x+2$ is divided by $x-2$.'

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'Find the general term of the sequence that represents the sum from the first term to the nth term as Sn'

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'Find the general term of the sequence {an} determined by the following conditions. a1=1, an+1=an+n^2'

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'For x=3+2i and y=3-2i, find the values of x+y, xy, and x^2+y^2, respectively.'

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'Demonstrate the general methods for addition, subtraction, and multiplication of complex numbers.'

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'Express the following expressions in the form r*sin(θ+α), where r>0, -π<α≤π.'

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'Find the sum S of an arithmetic series with the first term 3, common difference -5, up to the 11th term.'

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'When the polynomial P(x)=3x^{3}-ax+b is divided by x-2, the remainder is 24, and when divided by x+2, the remainder is -16. Find the values of constants a, b.'

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'Master polynomial division and conquer example 7!'

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'For the polynomials A and B, find the quotient and remainder when A is divided by B.\n(1) A=2 x^{3}+1-4 x, B=3-2 x+x^{2}\n(2) A=3 x^{3}+2 x^{2}+5, B=3 x+5'

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'Find the general term of the sequence {an} determined by the following conditions. (1) a1=-1, an+1=an+4 n-1 (2) a1=1, an+1=an+n^{2} (3) a1=4, an+1=an+5^{n}'

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'Find the values of real numbers x and y that satisfy the equations.'

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'Using long division, find the quotient and remainder when the following polynomial A is divided by the linear equation B.'

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'Find the values of real numbers x and y that satisfy the equation (2+i)(x+yi)=3-2i.'

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'When the polynomial P(x)=x^{3}-x^{2}+ax-4 is divided by x+1 with a remainder of -2, find the value of the constant a.'

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'Find the quotient Q and remainder R when the polynomial A is divided by B. Also, express the result in the form A=BQ+R. \n(1) A=4x^{3}-3x-9, B=2x+3 \n(2) A=1+2x^{2}+2x^{3}, B=1+2x'

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'Find the sum \\( \\sum_{k=1}^{n} k\\left(k^{2}-1\\right) \\).'

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'The slope of the line 2x+5y=3 is -2/5. Find the slope of a line perpendicular to this line, and derive its equation.'

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'Standard 56: Determining the remainder of polynomial division'

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'(4) Addition and subtraction\nWhen adding or subtracting fractions with different denominators, convert them to have the same denominator before performing the calculation.\nBringing the denominators of two or more fractions to be the same is called finding a common denominator.\nExamples:\n\\[\egin{array}{l}\\frac{x+2}{x+3}+\\frac{x+5}{x+3}=\\frac{(x+2)+(x+5)}{x+3}=\\frac{2x+7}{x+3}\\\\\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\\\\\frac{x+3}{x+5}-\\frac{1}{x+4}=\\frac{(x+3)(x+4)-1 \\cdot(x+5)}{(x+5)(x+4)}=\\frac{x^{2}+6x+7}{(x+5)(x+4)}\\quad\\frac{3}{5}-\\frac{1}{4}=\\frac{3 \\cdot 4-1 \\cdot 5}{5 \\cdot 4}=\\frac{7}{20}\\\\\\end{array}\\]'

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'Express the following expressions using ∑ notation.'

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'Lesson 33: Addition, Subtraction, and Multiplication of Complex Numbers'

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'(1) Find the remainder when the polynomial P(x) = 2x^3 - 3x + 1 is divided by the following linear expressions:\n(A) x-1\n(B) 2x+1\n(2) If the remainder is 17 when the polynomial P(x) = 1/2x^3 + ax + a^2 - 20 is divided by x-4, find the value of the constant a.\n(3) Given that the remainder is -5 when the polynomial P(x) = x^3 + ax^2 + x + b is divided by x+2, and the remainder is 20 when divided by x-3, find the values of the constants a and b.'

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'For building block C, choose an appropriate expression that represents the distance that building block B moved, and provide the symbols.'

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'Choose the correct answer from the following options.'

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'172 (1) \\( \\pi\\left(-r^{2} \\cos r+2 r \\sin r+2 \\cos r-2\\right) \\)\n(2) \ \\frac{1}{\\pi r^{2} \\sin r} \'

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'In the complex plane, there is a triangle OAB with three points O, A, and B as vertices. Here, O is the origin. Let P be the circumcenter of triangle OAB. If the complex numbers represented by points A, B, and P are α, β, and z respectively, and it is given that αβ=z, then determine the conditions that α must satisfy, and illustrate the shape described by point A(α) on the complex plane.'

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'Express the following expressions using z and its conjugate z-bar: (1) a (2) b (3) a-b (4) a^2-b^2'

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'(1) \\ 2 \\ sqrt{2} \\ times\\left(cos \\ frac{7}{4} \\ pi+i \\ sin \\ frac{7}{4} \\ pi\\right)'

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'Let z and w be complex numbers such that |z|=2,|w|=5. If the real part of z \x08ar{w} is 3, find the value of |z-w|.'

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'Find the general term of the following system of linear recurrence equations: \\( \\left\\{ \egin{array}{l}a_{n+1}=p a_{n}+q b_{n} \\\\ b_{n+1}=r a_{n}+s b_{n}\\end{array} \\quad(prs \\neq 0) \\)'

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'(1) \\ 2 \\cos \\frac{5}{12}\\pi\\left(\\cos \\frac{7}{12}\\pi+i\\sin \\frac{7}{12}\\pi\\right)'

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'Let t be a real number. For vectors a=(2,1) and b=(3,4), |a+tb| takes its minimum value when t = '

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'For the given vectors $\\ vec{a} = (11,23)$ and $\\ vec{b} = (-2,-3)$, what is the value of the real number t that minimizes $| \\ vec{a} + t \\ vec{b} |$? Furthermore, find the minimum value of $10| \\ vec{a} + t \\ vec{b} |$.'

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'A vector is a directed line segment starting from point A to point B. A vector is a quantity defined only by its direction and magnitude, denoted as a = AB. Answer the following questions: 1. Find the length of the directed line segment representing the magnitude of vector a, |a| a. 2. Describe the direction of ka when k > 0. 3. Explain the properties of the zero vector 0.'

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'Transform the following expressions into forms that do not include sqrt.'

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'Roll a die n times, multiply the outcomes, and get X. Let Y_k be the outcome on the k-th roll, then X = Y_1 Y_2 ... Y_n. Find the probability p_n that X is divisible by 3. [Kyoto University]'

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'(1) Find the value of the following expressions when \ 0^{\\circ} \\leqq \\theta \\leqq 180^{\\circ}, \\sin \\theta+\\cos \\theta=\\frac{1}{2} \:\\n(1) \ \\sin \\theta \\cos \\theta \\\n(2) \ \\sin ^{3} \\theta+\\cos ^{3} \\theta \\\n(3) \ \\sin \\theta-\\cos \\theta \\\n[Osaka University] Basic 25,113'

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'3\n\\[\n\egin{aligned}\n& \\text { (1) } A+B \\\n= \\left(5 x^{3}-2 x^{2}+3 x+4\\right)+\\left(3 x^{3}-5 x^{2}+3\\right) \\\n= (5+3) x^{3}+(-2-5) x^{2}+3 x+(4+3) \\\n= 8 x^{3}-7 x^{2}+3 x+7\n\\end{aligned}\n\\]\n(2)\n\\[\n\egin{aligned}\n& A-B \\\n= \\left(5 x^{3}-2 x^{2}+3 x+4\\right)-\\left(3 x^{3}-5 x^{2}+3\\right) \\\n= 5 x^{3}-2 x^{2}+3 x+4-3 x^{3}+5 x^{2}-3 \\\n= (5-3) x^{3}+(-2+5) x^{2}+3 x+(4-3) \\\n= 2 x^{3}+3 x^{2}+3 x+1\n\\end{aligned}\n\\]'

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'(1) \\( \\left(-2 x^{2} y\\right)^{2}(2 x-3 y) \\)'

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'In short-distance running in track and field, the time it takes to run 100 meters (hereinafter referred to as time) is related to the distance traveled per step (hereinafter referred to as stride) and the number of steps per second (hereinafter referred to as pitch). Stride and pitch are given by the following equations.'

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'By adding instead of subtracting the expression -2x^{2}+5x-3 from a certain polynomial, the answer turned out to be -4x^{2}+13x-6. Find the correct answer.'

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'Find the total number of ways to arrange the numbers when there are 68 numbers, with 1 appearing 3 times, 2 appearing 3 times, and 3 appearing 2 times.'

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'Permutation of numbers (based on the condition of number ordering)'

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'Consider the number of ways to divide a group of people into groups, ensuring that each group has at least one person.'

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'Instead of subtracting from a polynomial (-2x^2+5x-3), the expression was mistakenly added, resulting in -4x^2+13x-6. Find the correct answer.'

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'Find the expression to subtract from (2) to get 8x^2-5xy+5y^2.'

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'x = \\frac{-(-\\sqrt{2}) \\pm \\sqrt{(-\\sqrt{2})^{2}-4 \\cdot 1 \\cdot(-4)}}{2 \\cdot 1}'

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'Combine like terms in the given polynomial. Identify the degree and constant term when focusing on the characters in [ ].'

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'After learning about dealing with expressions involving absolute values, necessary and sufficient conditions, please attempt the following problem.'

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'Calculate the following expressions:\n\n(1) \\sqrt{27}-5 \\sqrt{50}+3 \\sqrt{8}-4 \\sqrt{75}\n\n(2) (\\sqrt{11}-\\sqrt{3})(\\sqrt{11}+\\sqrt{3})\n\n(3) (2 \\sqrt{2}-\\sqrt{27})^{2}\n\n(4) (3+4 \\sqrt{2})(2-5 \\sqrt{2})\n\n(5) (\\sqrt{10}-2 \\sqrt{5})(\\sqrt{5}+\\sqrt{10})\n\n(6) (1+\\sqrt{3})^{3}'

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'Perform the following polynomial calculations according to the rules of addition, subtraction, and multiplication of polynomials:'

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'How many ways are there to divide 8 apples into four bags labeled A, B, C, and D, with the possibility of some bags being empty?'

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"Questions involving addition, subtraction, and multiplication of polynomials often arise, where we are required to add, subtract, or find the product of different polynomials. Let's deepen our understanding through an example."

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'Determine the coefficients and degrees of the following monomials. Also, analyze the significance of the letters within the square brackets.'

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'Translate the given text into multiple languages.'

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'Calculate the following expressions when A=2x^{3} +3x^{2}+5, B=x^{3}+3x+3, C=-x^{3} -15x^{2} + 7x. (1) 4A + 3(A - 3B - C) - 2(A - 2C) (2) 4A - 2{B - 2(C - A)}'

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'Simplify the following expression. Where n is a natural number. (1) 2(-a b)^{n}+3(-1)^{n+1} a^{n} b^{n}+a^{n}(-b)^{n}'

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'Chapter 2 Real Numbers, Linear Inequalities: Calculations involving expressions with square roots of 5'

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'Calculate the sum and difference of polynomials A and B.'

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'Tell the degree and coefficient of the following monomials. Also, when focusing on the letter inside the square brackets, tell the degree and coefficient of that letter.'

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'Find the sum A + B and difference A - B of polynomials A and B.'

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'Standard 5: Addition and Subtraction of Polynomials (2)'

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'Chapter 1: Calculation of Equations - 1 Polynomial Addition and Subtraction'

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'Perform the following calculations. \n(1) \\( \\left(5 x^{3}+3 x-2 x^{2}-4\\right)+\\left(3 x^{3}-3 x^{2}+5\\right) \\)\n(2) \\( \\left(2 x^{2}-7 x y+3 y^{2}\\right)-\\left(3 x^{2}+2 x y-y^{2}\\right) \\)'

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"Explain the basic properties of sets A and B, and demonstrate De Morgan's Law."

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'When $A=2 x+y+z, B=x+2 y+2 z, C=x-2 y-3 z$, calculate the following expressions. (1) $3 A-2 C$ (2) $A-2(B-C)-3 C$'

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'(4) Multiply each side of -1<y<3 by -2 to get -1*(-2)>y*(-2)>3*(-2)'

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'Perform multiplication of monomials. Use the law of exponents.'

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'Calculate the sum and difference of the given polynomials A and B.'

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'Calculate the following expression: (8)(√6 + 2)(√3 - √2)'

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'Using the following multiplication rules, correct the incorrect statements in the table.'

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"If five fragrances are distinguished sequentially as having the same marks for the 1st and 4th, and the 2nd, 3rd, and 5th, it is represented in the pattern on the right. This pattern is named 'Suma'. There are a total of 52 patterns representing the distinction of five fragrances. Each of them is associated with the chapter names of The Tale of Genji, excluding 'Kiritsubo' and 'Yumeutsutsu'. These are known as Genji's incense patterns. Please consider how many patterns exist when there are two types of fragrances. How many patterns are there when the 5 fragrances are divided into 3 and 2 respectively? Also, consider the scenario of dividing them into 4 and 1."

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'Calculate the following expression: (2) 2√50 - 5√18 + 3√32'

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'1) In how many ways can 8 different juices be divided between two people, A and B, ensuring that each person receives at least 1 juice?\n2) In how many ways can 8 different juices be divided into 2 groups?'

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'Calculate the following expression: (7)(√20 + √3)(√5 - √27)'

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'Calculate expressions containing square roots. Calculate the following expression: \ \\sqrt{4} + \\sqrt{9} \'

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'By adding the expression B=2x^2-2xy+y^2 to equation A, but mistakenly subtracting B, the incorrect answer x^2+xy+y^2 was obtained. Find the correct answer.'

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'Perform addition and subtraction of polynomials. For example, calculate the following problem.'

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Join our Discord#### Q.82

'Calculate the following expression: (3)√2(√3 + √50) - √3(1 - √75)'

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Join our Discord#### Q.83

'Simplify the expression using the laws of indices.'

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Join our Discord#### Q.84

'When throwing three dice of different sizes simultaneously, how many ways are there for all dice to show odd numbers?'

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'Calculate the following equations in vertical format.'

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'(1) How many ways are there to put 10 people into 2 rooms, A or B? Allowing all of them to be in one room is also acceptable.\n(2) How many ways are there to divide 10 people into 2 groups, A and B?\n(3) How many ways are there to divide 10 people into 2 groups?'

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'Calculate the following expression: (1) 3√3 - 6√3 + 5√3'

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'Explain the calculation of the following expression: 500x z^3 times \\frac{1}{4} x^2 y^4 times \\frac{8}{125} x^3 z^3'

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'Considering 3 women as one group, the total number of circular permutations of 10 men and one group of women is (11-1)!=10! ways. In any case, the arrangement of the 3 women has 3! ways. Therefore, the required probability is (10!×3!)/(12!)=3×2×1 / 12×11=1/22.'

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'There is 1 red jade, 2 blue jades, 2 yellow jades, and 2 white jades.\n(1) How many ways are there to arrange all 7 jades in a circular form?\n(2) When threading all 7 jades and making bracelets, how many different bracelets can be created.'

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Join our Discord#### Q.93

'Comparison of powers and power roots: Compare the magnitude of the following powers and power roots.'

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Join our Discord#### Q.94

'Find S_n when n is even and odd: S_{n}=\x0crac{1}{2} n(n+2) when n is even, S_{n}=-\x0crac{1}{2}(n+1)^{2} when n is odd'

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'For a geometric sequence \ \\left\\{a_{n}\\right\\} \ with a common ratio of 2 and initial term of 1, find the sum \ \\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\frac{1}{a_{3}}+\\cdots \\cdots+\\frac{1}{a_{n}} \.'

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'For solution (II), lowering the degrees as done in solution (IV) is also a good method.'

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'(4)\n\\[\n\egin{aligned}\n\oldsymbol{y}^{\\prime} & =-\\left(x^{3}\\right)^{\\prime}+5\\left(x^{2}\\right)^{\\prime}-4(x)^{\\prime}+(1)^{\\prime} \\\\\n& =-1 \\cdot 3 x^{2}+5 \\cdot 2 x-4 \\cdot 1 \\\\\n& =-3 x^{2}+10 x-4\n\\end{aligned}\n\\]'

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'Calculate the following expressions. (1) \ \\frac{x^{2}+4 x+5}{x+3}-\\frac{x^{2}+5 x+6}{x+4} \ (2) \ \\frac{x+2}{x}-\\frac{x+3}{x+1}-\\frac{x-5}{x-3}+\\frac{x-6}{x-4} \'

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"(2) $y'=3\\left(x^{2}\\right)'-6(x)'(2)'=3 \\cdot 2 x-6 \\cdot 1$"

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'(1) Perform the following calculations:\n\nHere, $\\frac{1}{(x-3)(x-1)}+\\frac{1}{(x-1)(x+1)}+\\frac{1}{(x+1)(x+3)}$\n\n(2) Perform the following calculation:\n$\\frac{1}{a^{2}-a}+\\frac{1}{a^{2}+a}+\\frac{1}{a^{2}+3a+2}$'

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'Determine the values of constants a, b, c, d, e so that the first expression is divisible by the second expression.'

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'(4) Calculate $\\left(a^{-\\frac{1}{3}} b^{\\frac{1}{2}}\\right)^{2} \\times a^{\\frac{5}{3}}$'

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'Find the general term of the sequence {an} determined by the following conditions.\n(1) a1=6, an+1=3an-8'

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'Calculate the following expressions and find the solutions:\n13. (1) 1\n(2) \x0crac{x+4}{x+1}\n(3) 2 a-3\n(4) 1'

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'Find the value of the following expressions:\n15. (1) \\frac{2 x}{1+x^{2}}\n(2) -x+2'

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'Prove: (1) Prove that when a: b=c: d, the equation (a+c)(d+f)=(a+c+e)(b+d+f) holds true. (2) Prove that when a/b=c/d=e/f, the equation (a+c)/(b+d)=(a+c+e)/(b+d+f) holds true.'

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'Find the number which, when squared, results in 8i.'

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'\\\frac{x+1}{3 x^{2}-2 x-1}+\\frac{2 x+1}{3 x^{2}+4 x+1}\'

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'What should I do? Hanako: Since the remainder when dividing P(x) by (x-1)^{2} is 2x+3, we can express it as s x^{2}+t x+u=□. (i) Choose one expression that fits into the blank from the following options. (0) s x^{2}+5 (1) s x^{2}+2 s x+3 (2) s(x-1)^{2} (3) s(x-1)^{2}+5 (4) s(x-1)^{2}+2 x+3 (5) s\\left(x^{2}+2 x+3\\right) (ii) Determine the values of s, t, u, such that s=□, t=□, u=□. Fill in the correct numbers.'

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'Find the remainder when the given polynomial is divided by the specified linear equation. (1) $x^{3}+2 x-1 \\quad[x-3]$ (2) $2 x^{3}-x^{2}+3 x+1 \\quad[x+1]$ (3) $2 x^{4}+3 x^{3}-3 x^{2}-2 x+3 \\quad[x-1]$ (4) $x^{3}-5 x^{2}+4 \\quad[2 x-1]$'

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"If a < 0, b < 0, let a = -a', b = -b', where a' > 0, b' > 0, then"

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'Express the following expressions in the form of r\\sin (\\theta+\\alpha), where r>0, -\\pi<\\alpha\\leq\\\\pi.'

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'Compute the following calculations. Note that a>0, b>0.'

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'Find the values of b_n, c_n, and a_n using the following series of steps: (1) $b_{n+1}=\x0crac{n+2}{n} b_{n}$ (2) $c_{n}=\x0crac{1}{2}$ (3) $a_{n}=\x0crac{1}{4} n^{2}+\x0crac{1}{4} n+\x0crac{1}{2}$'

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'Find the sum of a geometric series with initial term 1 and common ratio 2, expressed as log₂(a₁) + log₂(a₂) + ... + log₂(aₙ).'

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'When a polynomial A is divided by the polynomial 2x^{2}-1, and the quotient is 2x-1 with a remainder of x-2, find A.'

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'(3) Using the distributive law $(a+b i)(c+d i)=a c+(a d+b c) i+b d i^{2}$'

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'When the polynomial $x^{3}+a x^{2}+b x-a$ is divided by $x^{2}+x+1$ and the remainder is $-x+3$, find the values of constants $a, b$.'

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'Find the quotient and remainder when the polynomial $x^3-2x^2-45x-40$ is divided by the polynomial $x-8$.'

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'Determine whether the following equation is an identity.'

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'Find the sum and product of the given complex number and its conjugate for each of the following:'

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'\\frac{2 a^{2}-a-3}{3 a-1} \\div \\frac{3 a^{2}+2 a-1}{9 a^{2}-6 a+1}'

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Join our Discord#### Q.28

'Let two real numbers a, b be positive. Also, let i be the imaginary unit.'

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'It is not possible to find the minimum value from mathematical Im. Therefore, for functions f(x) = 2x + 1 + 3/(x+1) (x > 0) and g(x) = 2 sqrt((2x+1) * 3/(x+1)) (x > 0), the relationship between the graphs of y = f(x) and y = g(x) is as shown in the diagram on the right, sharing the point (1/2, 4), but this point of intersection is not the point where y = f(x) is minimum. The specific value can be obtained in (2), where y = f(x) is minimum at x = 1/2. In (2), 2(a + (* *)) * 3/(a+1) becomes a constant when (a + (* *)) = a + 1, that is, (* *) = 11. Since a > 0, 2(a+1) > 0, 3/(a+1) > 0, by the inequality of arithmetic mean and geometric mean, 2a + 1 + 3/(a+1) = 2(a+1) + 3/(a+1) - 1 >= 2 sqrt(2(a+1) * 3/(a+1)) - 1 = 2 sqrt(6) - 1. Equality holds when 2(a+1) = 3/(a+1), and at this point (a+1)^2 = 3/2, a + 1 > 0 gives a + 1 = sqrt(6)/2, hence a = sqrt(6)/2 - 1 + sqrt(6)/2, the minimum value at this point is 2 sqrt(6) - 1.'

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Join our Discord#### Q.33

'Find the following expressions:\n17. (1) \\frac{3}{x(x+1)}\n(2) -\\frac{2(2 x+7)}{(x+2)(x+3)(x+4)(x+5)}'

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Join our Discord#### Q.35

'Rewrite the following angles in terms of radians and vice versa.'

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Join our Discord#### Q.36

'Calculate the following expression: (x+2)/(x^2+7x+12) - (x+4)/(x^2+5x+6) - (x^2+3x)/((x+2)(x^2+7x+12))'

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'The maximum profit at that time is a · 50 + 3 · 20 = 50a + 60 (million yen)'

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'Toss a coin once, getting 1 point for heads and 2 points for tails. Repeat this experiment n times, divide the sum of points by 3, the probability of the remainder being 0 is a_{n}, the probability of the remainder being 1 is b_{n}, and the probability of the remainder being 2 is c_{n}. (1) Find a_{1}, b_{1}, c_{1}. (2) Express a_{n+1} in terms of b_{n} and c_{n}. (3) Express a_{n+1} in terms of a_{n}. (4) Express a_{n} in terms of n.'

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'When simultaneously tossing 2 coins of 50 yen, 4 coins of 100 yen, and 1 coin of 500 yen, calculate the expected value and standard deviation of the total amount of coins facing up.'

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'Prove that the sequence where the nth term is 5n+1 is an arithmetic sequence, and find its first term and common difference.'

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'Basic Example 19: Identity of Fractional Equations (Partial Fraction Decomposition)'

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'Basic Example 55 Evaluate a higher degree polynomial using division\nFor P(x)=x^{3}+3 x^{2}+x+2, answer the following questions:\n(1) Prove that x=-1+i satisfies x^{2}+2 x+2=0.\n(2) Find the quotient and remainder when P(x) is divided by x^{2}+2 x+2.\n(3) Find the value of P(-1+i).'

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'What is the remainder when the polynomial P(x) is divided by the linear expression ax+b?'

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'Find the quotient and remainder when polynomial A is divided by polynomial B for the following values of x:'

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'Prove that the equation a³(b-c) + b³(c-a) + c³(a-b) = 0 holds true when a+b+c=0.'

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'For this division, the following equation holds.'

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'Express the relative sizes of the following sets of numbers using inequality symbols.'

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'Show the addition and subtraction of complex number z = a + bi.'

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'Choose 3 cards out of 9 cards with numbers from 1 to 9 written on them and arrange them to form a 3-digit number.'

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'(2) Relationship between size and sign of difference 5. a>b ⇔ a-b>0 6. a<b ⇔ a-b<0'

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'Find the general term of the sequence {a_n} determined by the following conditions.'

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'Perform the following calculation: (1) (x+2)/(x^2+7x+12) - (x+4)/(x^2+5x+6) - (x^2+3x)/((x+2)(x^2+7x+12))'

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'Please solve the problem using the following exponent rule: Exponent Rule: a^{r} a^{s}=a^{r+s} Specifically, when a is 3, r is 2, and s is 4, find the value of a^{r+s}.'

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'Simplify the following fractions and express them in simplest form.'

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'Company X, which operates a bike-sharing system, is planning to set up two locations, A and B, in a town. Each location will have a large number of rental bikes, and users are allowed to borrow from and return to either location. Every day, all the bikes at locations A and B are rented out only once and returned to either location on the same day. It is assumed that the ratio of bikes returned to each location remains constant. Specifically, 70% of bikes borrowed from A are returned to A, and 30% are returned to B. For bikes borrowed from B, 20% are returned to A, and 80% are returned to B. Let an and bn be the ratios of the number of bikes at locations A and B, respectively, to the total number of bikes after the end of the nth day. If the initial proportions of bikes at A and B are 20% and 80%, respectively, answer the following question: (1) Find an.'

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'\\(\\frac{(a+1)^{2}}{a^{2}-1} \\times \\frac{a^{3}-1}{a^{3}+1} \\div \\frac{a^{2}+a+1}{a^{2}-a+1}\\)'

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'Find the value of the following expression: \\(\\frac{1}{2}(\\sqrt{2 n+1}-1) \\)'

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'Express the relative sizes of each set of numbers using inequalities.'

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'Using long division, find the quotient and remainder when polynomial A is divided by polynomial B. (1) A=x^{3}+2 x^{2}-x-3, B=x+3 (2) A=2 x^{3}+x^{2}+x-2, B=2 x-1'

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'Organize the like terms in the following polynomials and determine the degrees and constant terms of the letters inside the square brackets in polynomials (2) and (3).'

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'Simplify the following expressions by removing the double square root.'

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'Let 12 be a real number, and b be a positive constant. Find the minimum value m of the function f(x) = x^{2} + 2(ax + b|x|). Furthermore, plot a graph of m with a on the horizontal axis and m on the vertical axis as the value of a changes.'

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Join our Discord#### Q.69

'Basic matters\n3 square roots\n(1) The number whose square is equal to a is called the square root of a.\n(2) Properties 1. When a ≥ 0, (√a)² = a, (-√a)² = a, √a ≥ 0\n2. When a ≥ 0, √(a²) = a; when a < 0, √(a²) = -a, that is, √(a²) = |a|\n(3) Formulas When a > 0, b > 0, k > 0\n3. √a * √b = √(a * b); 4. (√a) / (√b) = √(a / b); 5. √(k² * a) = k * √a\n\nRationalizing the denominator Transforming an expression containing a radical in the denominator into one that does not contain a radical is called rationalizing the denominator.'

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'Please calculate the following examples using the laws of exponents.'

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'Translate the given text into multiple languages.'

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'Please provide an example of polynomial addition and subtraction.'

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Join our Discord#### Q.76

'Simplify the following expressions by removing the double square roots.'

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'(2) When \-2<x<\\frac{3}{4}\, \ x+2>0,\\; 4 x-3<0 \, therefore, the expression \\(\\sqrt{(x+2)^{2}}-\\sqrt{(4 x-3)^{2}}=|x+2|-|4 x-3|= (x+2)-\\{-(4 x-3)\\}\\)'

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Join our Discord#### Q.79

'Remove the square roots in the following expression and simplify: $\\sqrt{x^{2}+4 x+4}-\\sqrt{16 x^{2}-24 x+9}$ (where $-2<x<\\frac{3}{4}$).'

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'Let a = b = c be real numbers, and define A, B, C as A = a + b + c, B = a^{2} + b^{2} + c^{2}, C = a^{3} + b^{3} + c^{3}. Express abc in terms of A, B, C.'

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'(1) \ \\frac{3 \\sqrt{2}}{2 \\sqrt{3}}-\\frac{\\sqrt{3}}{3 \\sqrt{2}}+\\frac{1}{2 \\sqrt{6}} \'

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'Calculate the following mathematical expressions.'

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'From the sine theorem, $\\frac{C}{\\sin C}=2 R$, so\n\ \egin{aligned} c & =2 R \\sin C=2 \\cdot 4 \\sin 120^{\\circ} \\\\ & =2 \\cdot 4 \\cdot \\frac{\\sqrt{3}}{2}=4 \\sqrt{3} \\end{aligned} \\]\nFrom the sine theorem\n\\[ \\frac{a}{\\sin A}=\\frac{b}{\\sin B}=2 R \\]\nTherefore\n\\[ \egin{aligned} b & =\\sin B \\cdot \\frac{a}{\\sin A} \\\\ & =\\sin 60^{\\circ} \\cdot \\frac{2}{\\sin 45^{\\circ}} \\\\ & =\\frac{\\sqrt{3}}{2} \\cdot \\frac{2}{\\frac{1}{\\sqrt{2}}} \\\\ & =\\sqrt{3} \\cdot \\sqrt{2}=\\sqrt{6} \\\\ R & =\\frac{1}{2} \\cdot \\frac{a}{\\sin A}=\\frac{1}{2} \\cdot \\frac{2}{\\sin 45^{\\circ}}=\\sqrt{2} \\end{aligned} \'

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'Find the equation of a parabola that is obtained by translating 2 units in the x-axis direction and -1 unit in the y-axis direction, such that it overlaps with the parabola y=-2 x^{2}+3.'

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'Find the value of PR(2) sin 160° cos 70° + cos 20° sin 70°.'

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'Addition, subtraction, and multiplication of polynomials'

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'Simplify the following equations by removing the square roots.'

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'By mistakenly adding the polynomial instead of subtracting it, the answer became -4x^2 + 13x - 6. Find the correct answer.'

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'In part (1) of Basic Example 13, think about how to consolidate and replace common expressions.'

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'List the basic laws of polynomial addition and multiplication.'

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'Using (3)(2), Taro decided to determine the price of one okonomiyaki so that the profit is maximized. Find the value of x that maximizes the profit, and calculate the profit at that point.'

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'Addition and subtraction of polynomials\nThe sum A+B is obtained by adding all the terms of A and B, and if there are similar terms, they are combined and simplified.\nThe difference A-B is considered as A+(-B), where the sign of each term in B is changed and added to A.\nVertical calculation\nAs shown on the right, it is also acceptable to align similar terms and perform vertical calculations. In this case, leave space for missing degree terms.'

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'In basic examples 6 and 12, a method of combining common expressions and then proceeding with calculations is introduced. Please explain the method you would use to advance the solution.'

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'(1) Show the expressions for replacing x with x+1 and y with y-2. (2) For the function f(x) = -2x^2 + 1, show the resulting function.'

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'The following calculation is incorrect. List all incorrect equalities and explain the reason for considering them incorrect.'

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'Subtracting -2x^2 + 5x - 3 from a certain polynomial, but mistakenly adding this expression resulted in -4x^2 + 13x - 6. Find the correct answer.'

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'(1) When $x=3$, $f(3)=-2 \\cdot 3+1=-6+1=-5$. (2) When $x=0$, $f(0)=-2 \\cdot 0+1=1$'

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'Please explain the sum, difference, product, and quotient of two rational numbers.'

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'It is possible to create common factors through the manipulation of terms.'

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'Given x=\\\sqrt{2}+\\sqrt{3}\, find the values of x^{2}+\\frac{1}{x^{2}}, x^{4}+\\frac{1}{x^{4}}, x^{6}+\\frac{1}{x^{6}}. [Rikkyo University]Since x=\\\sqrt{2}+\\sqrt{3}\\n\\[\egin{aligned} \\frac{1}{x} &=\\frac{1}{\\sqrt{2}+\\sqrt{3}}=\\frac{1}{\\sqrt{3}+\\sqrt{2}}=\\frac{\\sqrt{3}-\\sqrt{2}}{(\\sqrt{3}+\\sqrt{2})(\\sqrt{3}-\\sqrt{2})} \n&=\\sqrt{3}-\\sqrt{2} \\text{and} \\, x+\\frac{1}{x}=(\\sqrt{2}+\\sqrt{3})+(\\sqrt{3}-\\sqrt{2})=2 \\sqrt{3} \\end{aligned}\\]\nTherefore, x+\\frac{1}{x}=(\\sqrt{2}+\\sqrt{3})+(\\sqrt{3}-\\sqrt{2})=2 \\sqrt{3} hence x^{2}+\\frac{1}{x^{2}}=\\left(x+\\frac{1}{x}\\right)^{2}-2\\ n\\[\egin{aligned}\n& =(2 \\sqrt{3})^{2}-2=10 \\ x^{4}+\\frac{1}{x^{4}} &=\\left(x^{2}+\\frac{1}{x^{2}}\\right)^{2}-2 \n&=10^{2}-2=98\n\\end{aligned}\\]'

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"Challenge Question\\nIn the 100m sprint event of 33 track and field competitions, the time it takes to run 100m (referred to as time) is related to the distance covered per step (referred to as stride) and the number of steps per second (referred to as pitch). The stride and pitch are given by the following equations. \\n\\[ \\n\egin{array}{l} \\n\\text{Stride} (m/step) = \\frac{100 (m)}{number of steps to cover 100m (steps)} \\n\\text{Pitch} (steps/second) = \\frac{number of steps to cover 100m (steps)}{time (seconds)} \\n\\end{array} \\n\\] \\nHowever, the number of steps to cover 100m can be a decimal as the last step may cross the finish line. Unless otherwise specified, units are omitted. \\nFor example, when the time is 10.81 and the number of steps is 48.5, the stride is approximately \\\frac{100}{48.5}\ which is about 2.06, and the pitch is approximately \\\frac{48.5}{10.81}\ which is about 4.49. \\nWhen answering in decimal form, round to the next digit of the specified precision. \\n(1) Let's denote the stride as x and the pitch as z. The pitch is the number of steps per second, and the stride is the distance covered per step, so the average speed per second, denoted by A (m/s), is given by x and z. \\nTherefore, the relationship between time and stride, pitch can be expressed as \\n\\text{Time} = \\frac{100}{\\square A}\\nand the time becomes best when A is maximized. However, becoming best means that the value of time becomes smaller. \\nChoose one from the following options (0) \x+z\, (1) \z-x\, (2) \xz\, (3) \\\frac{x+z}{2}\, (4) \\\frac{z-x}{2}\, (5) \\\frac{xz}{2}\"

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'Perform the following calculations:\n(1) A+B\n(2) A-B\nwhere A=5x^3-2x^2+3x+4, B=3x^3-5x^2+3'

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'When the values of two numbers a and b are in the range of -2 ≤ a ≤ 1 and 0 < b < 3, find the range of possible values for 1/2a - 3b.'

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'Find the vector x that satisfies the following equations and express it in terms of vector a:\n(1) 4x - a = 3x + 2b\n(2) 2(x - 3a) + 3(x - 2b) = 0'

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'Three points A, B, and C are collinear if and only if AC=kAB, where k is a real number.'

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'Given three distinct complex numbers \ \\alpha, \eta, \\gamma \, if the equation \\( \\sqrt{3} \\gamma-i \eta=(\\sqrt{3}-i) \\alpha \\) holds, answer the following questions.'

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'(2) If complex numbers $\\alpha, \eta, \\gamma, \\delta$ satisfy $\\alpha+\eta+\\gamma+\\delta=0$ and $|\\alpha|=|\eta|=|\\gamma|=|\\delta|=1$, find the value of $|\\alpha-\eta|^2+|\\alpha-\\gamma|^2+|\\alpha-\\delta|^2$.'

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'(g∘f)(x) = g(f(x)) = 2f(x) - 1 = 2(x + 2) - 1 = 2x + 3 (f∘g)(x) = f(g(x)) = g(x) + 2 = (2x - 1) + 2 = 2x + 1'

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'Find the minimum value of $c\\sqrt{\\frac{a^{2}+b^{2}}{a^{2}+b^{2}+4 c^{2}}}$ when $s=\\frac{b^{2} c^{2}}{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}}$, $t=\\frac{c^{2} a^{2}}{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}}$, and $r=\\frac{4 c^{2}}{a^{2}+b^{2}+4 c^{2}}$.'

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'Exercise problem solution 60 (1) \\log \eta-\\log \\alpha-\\frac{2(\eta-\\alpha)}{\\alpha+\eta}'

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'Find the solution for (1): Calculate (g∘f)(x), (f∘g)(x).\nProve (2): (h∘(g∘f))(x)=((h∘g)∘f)(x).\nf(x)=x+2, g(x)=2x-1, h(x)=-x^{2}.'

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'(2) - (1) multiplied by 3 yields (3), (4) yields z = \\frac{-x+1}{3}, z=-y+1'

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'Conditions for three points A, B, and C to be collinear and perpendicular conditions\nLet c be a real constant. Represent points A, B, and C as α=1+i, β=-i, γ=-2+ci.\n(1) Determine the value of c so that points A, B, and C are collinear.\n(2) Determine the value of c so that lines AB and AC are perpendicular.'

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'Practice problem 78\n(1) When \\( (1-\\cos \\theta-i \\sin \\theta)^{-1} \\) is represented as \ \\frac{1}{2}+\\frac{i}{a} \, find the value of the real number \ a \.\n(2) Using \\( (1-\\cos \\theta-i \\sin \\theta)\\left(1+\\cos \\theta+i \\sin \\theta+(\\cos \\theta+i \\sin \\theta)^{n}\\right)=1-(\\cos \\theta+i \\sin \\theta)^{n+1} \\) to show\n\\n\egin{\overlineray}{l}\n1+\\cos \\theta+\\cos 2 \\theta+\\cdots+\\cos n \\theta=\\frac{\\sin \\frac{n+1}{2} \\theta \\cos \\frac{n}{2} \\theta}{\\sin \\frac{1}{2} \\theta} \\text {, } \\\\\n\\sin \\theta+\\sin 2 \\theta+\\cdots+\\sin n \\theta=\\frac{\\sin \\frac{n+1}{2} \\theta \\sin \\frac{n}{2} \\theta}{\\sin \\frac{1}{2} \\theta} \\\\n\\end{\overlineray}\n\'

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'Translate the given text into multiple languages.'

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'Exercise problem answer 60 (2) t = \\sqrt{\\alpha \eta}'

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'Translate the given text into multiple languages.'

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'Let P(z) be a point on the line passing through two distinct points A(α) and B(β) on the unit circle. Prove the equation z+αβ𝑧¯=α+β.'

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'Express the practice z = sinα + i cosα (where 0 ≤ α < 2π) in polar form.'

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'Addition and subtraction of matrices, scalar multiplication'

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'Exercise Answer 63 (1) f(x)=x^{2}+\x0crac{1}{2} x, g(x)=\x0crac{1}{4} x^{2}-\x0crac{3}{2} x+4'

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'(1) \ \\frac{2}{3} a \\n(2) \ 3 a \\n(3) \ \\sqrt{3} a^{2} + \\frac{2}{3} \\pi a^{2} \'

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'Given \n\ \eta=a+b\\ i, z=x+y\\ i \, when substituting into \n\ \\overline{\eta} z+\eta \\overline{z}+c=0 \, we get\n\\[\n\egin{array}{l}\n(a-b\\ i)(x+y\\ i) \\\\\n(a+b\\ i)(x-y\\ i)+c=0 \\end{array} \n\\]\nSimplifying, we obtain the following equation:\n\\n2 a x+2 b y+c=0 \n\'

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'Addition, subtraction, and scalar multiplication of matrices'

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'(2) \ z + \\frac{1}{z} = -\\sqrt{2} \ Multiplying both sides by \ z \ and simplifying, we get\n\ z^{2} + \\sqrt{2} z + 1 = 0 \\nSolving for \ z \, we get \\( z = \\frac{-\\sqrt{2} \\pm \\sqrt{(\\sqrt{2})^{2} - 4 \\cdot 1 \\cdot 1}}{2 \\cdot 1} = \\frac{-\\sqrt{2} \\pm \\sqrt{2} i}{2} \\)\n\n\\( z^{12} + \\frac{1}{z^{12}} = (\\cos \\theta + i \\sin \\theta)^{12} + (\\cos \\theta + i \\sin \\theta)^{-12} \\)\n\\( = (\\cos 12 \\theta + i \\sin 12 \\theta) + \\{\\cos (-12 \\theta) + i \\sin (-12 \\theta)\\} \\)\n\\( = 2 \\cos 12 \\theta = 2 \\cos \\left\\{12 \\times ( \\pm \\frac{3}{4} \\pi) \\right\\} = 2 \\cos ( \\pm 9 \\pi) \\)\n\\( = 2 \\cos 9 \\pi = 2 \\times (-1) = -2 \\)'

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'For the matrices A=\\left(\egin{\overlineray}{ll}1 & 2 \\\\ 3 & 6\\end{\overlineray}\\right) and B=\\left(\egin{\overlineray}{ll}6 & x \\\\ y & z\\end{\overlineray}\\right), determine the values of $x, y, z$ to satisfy $AB=BA=O$.'

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'When $\\vec{a} = (-2,1)$ and $\\vec{b} = (3,-2)$, express the vector $5 \\vec{a} + 3 \\vec{b}$ in components. Also, find its magnitude.'

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'(Alternative Solution)\n\nWhen z is a real number, based on the quadratic equation (x + 1/x) ^ 2 = x^2 + 1 + 2, and (x + 1/x) ^ 3 = x^3 + 1/x^3 + 3(x + 1/x).\nConsidering z as a real number: (-√2) ^ 3 + 3√2 = 0.\nTherefore: √2.\n\nHence, the equation of z^6 + 1/z^6 equals (z^3 + 1/z^3)^2 - 2 = √2 - 2 = 0.\nTherefore, z^12 + 1/z^12 is equal to (z^6 + 1/z^6)^2 - 2 = 0^2 - 2 = -2.'

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'Considering the curve alpha z + beta, when point z moves on the circle with center at the origin O and radius 1, represented by w = (1-i)z - 2i. What kind of shape will the point w draw?'

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'For $k=0,1,2$, let $z$ be $z_{0}$, $z_{1}$, $z_{2}$'

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'Translate the given text into multiple languages.'

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'Exercise problem answer 64 (1) a=e-\x0crac{1}{2}, b=e-\x0crac{3}{2}'

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'Practice (2) Let \\( \\vec{a}=(2,3), \\vec{b}=(1,-1), \\vec{p}=\\vec{a}+k \\vec{b} \\). Find the maximum and minimum values of \ |\\vec{p}| \ when \ -2 \\leqq k \\leqq 2 \.'

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'Find the equations of the following lines:\n(1) 2y=2(x+1)\n(2) \\frac{\\sqrt{3} x}{12}+\\frac{\\sqrt{3} y}{4}=1\n(3) \\frac{-2 \\sqrt{5} x}{16}-\\frac{1 \\cdot y}{4}=1\n(4) 4 \\cdot 1 \\cdot x-5 \\cdot(-1) \\cdot y=-1'

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'Simplify the following expression: \\( \\frac{3x-5}{1-\\frac{1}{1-\\frac{1}{x+1}}} - \\frac{x(2x-3)}{1+\\frac{1}{1-\\frac{1}{x-1}}} \\) [(2) Musashi University]'

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'Given x = 1 + √2i, find the value of the following equation.'

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"For a polynomial f(x) in terms of x, if f(3) = 2 and f'(3) = 1, find the remainder when f(x) is divided by \\\\underline{201}(x-3)^{2}."

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'Divide the polynomial P(x) by x-1 to get a remainder of 5, and by x-2 to get a remainder of 7. Find the remainder when P(x) is divided by x^2-3x+2.'

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'Find the quotient and remainder when polynomial A is divided by polynomial B.'

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'Consider the line l: y=-2x, where point A(a, b) has a symmetric point B. Find the coordinates of point B in terms of a, b. Also, determine the equation of the path traced by point B as point A moves along the line y=x.'

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'Practice the following calculations: (3) 13 (1) $\\frac{x^{2}+2x+3}{x}-\\frac{x^{2}+3x+5}{x+1}$ (2) $\\frac{x+1}{x+2}-\\frac{x+2}{x+3}-\\frac{x+3}{x+4}+\\frac{x+4}{x+5}$'

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'Find the remainder when the polynomial P(x) is divided by the quadratic expression x^2+3x+2.'

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'Let n be a natural number greater than or equal to 2, and let i be the imaginary unit. When α=1+√3i and β=1-√3i, find the value of (√(β^{2}-4 β+8))/(α^{n+2}-α^{n+1}+2 α^{n}+4 α^{n-1}+α^{3}-2 α^{2}+5 α-2)^{3}.'

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'(2) Let the polynomial P(x) be divided by x-3 and (x+2)(x-1)(x-3) with remainders a and R(x) respectively. Given that the coefficient of x^2 in R(x) is 2. Furthermore, when P(x) is divided by (x+2)(x-1) the remainder is 4x-5. Find the value of a. [Similar to Hosei University]'

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'Prove that for any real numbers a, b, c satisfying a+b+c≠0 and abc≠0, the equation 1/a+1/b+1/c=1/(a+b+c) holds. In this case, prove that for any odd number n, the equation 1/a^n+1/b^n+1/c^n=1/(a+b+c)^n also holds.'

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'When the polynomial P(x) is divided by x^2-1, the remainder is 4x-3, and when divided by x^2-4, the remainder is 3x+5. Find the remainder when P(x) is divided by x^2+3x+2.'

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'Practice (1) Divide the polynomial $2x^{3} + ax^{2} + x + 1$ by the polynomial $x^{2} + x + 1$ to obtain a quotient of $bx - 1$ and a remainder of $R$. Find the values of the constants $a, b$ and $R$. Note that $R$ is a polynomial or a constant in terms of $x.'

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'When the polynomial P(x) is divided by x^{2}+5 x+4, the remainder is 2 x+4, and when divided by x^{2}+x-2, the remainder is -x+2. In this case, find the remainder when P(x) is divided by x^{2}+6 x+8.'

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'Calculate the following expressions:\n(1) \ \\frac{x^{2}+2 x+3}{x}-\\frac{x^{2}+3 x+5}{x+1} \\n(2) \ \\frac{x+1}{x+2}-\\frac{x+2}{x+3}-\\frac{x+3}{x+4}+\\frac{x+4}{x+5} \'

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'Find the remainder when P(x) is divided by (x-1)(x+2).'

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'(3) Long form 10 multiplication, division of fractions'

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'Find the remainder when a linear polynomial P(x) is divided by a linear factor (x-a).'

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'Find the quotient and remainder of polynomial A divided by polynomial B.'

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'Simplify the following expressions: (1) \ \\frac{\\frac{x^{4}-7 x^{2}+12}{x^{2}-x-6} \\times \\frac{2 x^{2}+7 x+3}{2 x+1}}{x^{2}+x-6} \ (2) \\( \\frac{3 x-5}{1-\\frac{1}{1-\\frac{1}{x+1}}}-\\frac{x(2 x-3)}{1+\\frac{1}{1-\\frac{1}{x-1}}} \\) [(2) Musashi University]'

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'Find the values of constants a and b for the function f(x)=ax^{n+1}+bx^n+1 such that the function is divisible by (x-1)^2.'

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'Division of polynomials and determination of polynomials'

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'Determine the range of real numbers for which the sequence {\\left(\\frac{2 x}{x^{2}+1}\\right)^{n}} converges. Also, find the limit of the sequence for those values of x.'

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'Let z ≠ 0. In the complex plane, when the point z and the point z^5 are symmetric with respect to the origin O, find the value of z. Also, in the complex plane, find the area of the polygon with the vertex corresponding to the calculated value of z.'

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'When the point P(x, y) makes one counterclockwise revolution around the circle with radius 1 centered at the origin, how many revolutions do points Q1(-y, x) and Q2(x^2 + y^2, 0) respectively make around the origin counterclockwise?'

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'Define functions f(n) and g(n) that take integer values n as follows: f(n)=1/2 n(n+1), g(n)=(-1)^{n}, define the composite function h(n)=g(f(n)). Furthermore, roll a six-sided dice 4 times, denote the outcomes as j, k, l, m, and let a=h(j), b=h(k), c=h(l), d=h(m), consider the function P(x)=a x^{3}-3 b x^{2}+3 c x-d.'

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"Alternative solution 1. Let's assume z=x+yi(x, y) is a real number"

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'(1) \\( \\sqrt{2}\\left(\\cos \\frac{5}{4} \\pi+i \\sin \\frac{5}{4} \\pi\\right) \\)\n(1) \\( \\cos \\left(\\frac{\\pi}{2}-\\alpha\\right)+i \\sin \\left(\\frac{\\pi}{2}-\\alpha\\right) \\)\n(2) \ z=1+i \'

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'(3) From the result of (2), we have aₙ₊₁ - 2/3 = -1/2 (aₙ - 2/3). Therefore, the sequence {aₙ - 2/3} with initial term a₁ - 2/3 = 1 - 2/3 = 1/3 and common ratio -1/2 is a geometric sequence. Thus, aₙ - 2/3 = 1/3 (-1/2)⁽ⁿ⁻¹⁾, hence aₙ = 1/3 (-1/2)⁽ⁿ⁻¹⁾ + 2/3. Therefore, limₙ→∞ aₙ = limₙ→∞ {1/3 (-1/2)⁽ⁿ⁻¹⁾ + 2/3} = 2/3'

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'(1) \\frac{2 t+1}{6 t^{2}} (2) -2 \\sqrt{1-t^{2}} (3) -\\frac{3 \\cos \\theta}{2 \\sin \\theta} (4) -\\frac{2}{3} \\tan \\theta'

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'Translate the given text into multiple languages.'

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'If g(x)=a x^{n+1}+b x^{n}+1 is divisible by (x-1)^{2}, express a and b in terms of n, where a and b are independent of x.'

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'Find the value of the following expressions:\n(1) (-√3 + i)^6\n(2) (1 + i) / 2)^{-14}'

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'Let {a_{n}} and {b_{n}} be sequences that satisfy the following relation.'

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'Which of the following complex numbers are real and which are purely imaginary? Assume that αβ̅ is not a real number.'

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'When a point z moves along a circle with radius 1 centered at the origin O, the point w represented by the following equation w=(1-i)z-2i will draw what kind of shape?'

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'Let PR α, β be complex numbers. (1) If α= |β|=1, α-β+1=0, find the values of α β and α/β+β/α. (2) If |α|=|β|=|α-β|=1, find the value of |2 β-α|.'

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'Translate the given text into multiple languages.'

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'\\( { }_{2} \\mathbf{e}=(1,0,0), \\vec{e}=(0,1,0), \\vec{e}=(0,0,1)\\) and \\( \\vec{a}=\\left(0, \\frac{1}{2}, \\frac{1}{2}\\right), \\vec{b}=\\left(\\frac{1}{2}, ０, \\frac{1}{2}\\right), \\vec{c}=\\left(\\frac{1}{2}, \\frac{1}{2}, ０\\right) \\) when \ \\vec{e}, \\vec{e}, \\vec{e} \\vec{e}_{3} \ are expressed using \ \\vec{a}, \\vec{b}, \\vec{c} \ respectively. Also, express \\( \\vec{d}=(3,4,5) \\) using \ \\vec{a}, \\vec{b}, \\vec{c} \.\\n[Kinki University]\\n(Latter half) Express \ \\vec{d} \ using \ \\overrightarrow{e_{1}}, \\overrightarrow{e_{2}}, \\overrightarrow{e_{3}} \ and substitute the results from the former half.'

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'Parallel condition of vectors: When vectors 𝐚≠0 ,𝐛≠0 , then 𝐚 // 𝐛⟺𝐛=k𝐚 where 𝑘 is a real number.'

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'When two non-parallel vectors \ \\vec{a}, \\vec{b} \ (where \ \\vec{a} \\neq \\overrightarrow{0}, \\vec{b} \\neq \\overrightarrow{0} \) satisfy \\( s(\\vec{a}+3 \\vec{b})+t(-2 \\vec{a}+\\vec{b})=-5 \\vec{a}-\\vec{b} \\), find the values of the real numbers \ s, t \.'

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'Solve the following equations using polar form: (1) $z^{6}=1$ (2) $z^{8}=1$'

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'Find the values of the constants a, b, and c when the composite function (g∘f)(x) = x is satisfied.'

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'Consider the following sequence of complex numbers.'

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'Express the following complex numbers in polar form. Where the range of the argument 𝜃 is 0 ≤ 𝜃 < 2𝜋. (1) 2(sin(𝜋/3) + cos(𝜋/3)) (2) 𝑧 = cos(12/7)𝜋 + i sin(12/7)𝜋 then -3𝑧'

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'By expressing 1+i and √3+i in polar form, find the values of cos(5/12π) and sin(5/12π) respectively.'

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"Let's denote c = a + tb for two vectors a = (11, -2) and b = (-4, 3). The real number t varies."

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'Let α=2+i and β=4+5i. Find the complex number γ representing the point β after rotating around point α by π/4.'

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'52 (1) \ \\frac{5}{2} x \\sqrt{x} \\n(2) \ -\\frac{2}{3 x \\sqrt[3]{x^{2}}} \\n(3) \ \\frac{4 x^{4}+3 x^{2}}{\\sqrt{1+x^{2}}} \'

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'(1) Prove that for any complex number z, z\ar{z}+\\alpha\ar{z}+\ar{\\alpha}z is a real number.\n(2) Show that for non-real complex number z where \\\overline{\\alpha} z\ is not real, \\\alpha\\overline{z}-\\overline{\\alpha}z\ is a pure imaginary number.'

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'When \ \\vec{x}=3 \\vec{a}-\\vec{b}+2 \\vec{c}, \\vec{y}=2 \\vec{a}+5 \\vec{b}-\\vec{c} \, express \\( 7(2 \\vec{x}-3 \\vec{y})-5(3 \\vec{x}-5 \\vec{y}) \\) in terms of \ \\vec{a} \, \ \\vec{b} \, \ \\vec{c} \.'

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'(1) \\( \\frac{3 x+2}{3 \\sqrt[3]{x(x+1)^{2}}} \\)\n(2) \ 2 x^{\\log x-1} \\log x \'

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'Given four distinct points O, A, B, and C not on the same plane, and for two points P, Q, if ⃗OP=⃗OA-⃗OB and ⃗OQ=-5⃗OC, find the values of real numbers k, l such that k⃗OP+⃗OQ=-3⃗OA+3⃗OB+l⃗OC holds true.'

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'Let the probability of event \ A \ occurring in 231 trials be \\( p(0<p<1) \\). If this experiment is conducted \ n \ times, let the probability of \ A \ occurring an odd number of times be denoted as \ a_{n} \.'

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'Express the following complex numbers in polar form. Where the argument θ satisfies 0 ≤ θ < 2π. (1) z = -cosα + i sinα (0 ≤ α < π) (2) z = sinα - i cosα (0 ≤ α < π/2)'

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'Translate the given text into multiple languages.'

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'13\n(2)\n\\[y^{\\prime} =(2 x-1)^{\\prime}(4 x+1)+(2 x-1)(4 x+1)^{\\prime} \\\\ =2(4 x+1)+(2 x-1) \\cdot 4 \\\\ =16 x-2\\]'

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'Transform the given rational function y=(6x+5)/(2x-1) into standard form y=k/(x-p)+q.'

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'For complex numbers z and w satisfying |z|=|w|=1 and zw≠1, prove that (z-w)/(1-zw) is real.'

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'When p = 1, a_{n}=2 n, when p ≠ 1, a_{n}=\\frac{2\\left(p^{n}-1\\right)}{p-1}; -1<p<1'

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'Let z = x + y, where x and y are real numbers, be treated as a real number equation. The biggest advantage of this approach is that it allows for easier calculation strategies as it can be thought of in familiar real numbers. However, calculations often become more complex. (Example 101, solution 1) In example 100, z = x+y with x and y as real numbers, solutions (1), (2), (4) can also be obtained, but compared to method 1, the amount of computation is higher.'

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'Let the partial sum from the first term to the nth term be S_n'

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'Find the composite function (f ∘ g)(x) of two functions f(x) and g(x). (1) Given f(x)=\\frac{x-1}{2x+3}, g(x)=\\frac{-x}{x+1}, find (f ∘ g)(x). (2) Let a, b be real numbers, and f(x)=\\frac{x+1}{ax+b}. Find the values of a, b that satisfy (f ∘ f)(x)=x. (3) Let a be a real number where a ≠ 0, and f(x)=\\frac{ax+1}{-ax}. Find the value of a that satisfies (f ∘(f ∘ f))(x)=x. Here, (f ∘(f ∘ f))(x) means f((f ∘ f)(x)). [Yamaguchi Hiroshi] Example 11'

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'Find the value of P = (-1 + √3 i) / 2)^n + ((-1 - √3 i) / 2)^n. Where n is a positive integer.'

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'Find the sum and difference of complex numbers α = a + bi, β = c + di.'

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'Translate the given text into multiple languages.'

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'Let a, b be constants such that 100<a<b. Define x_{n}= (a^{n}/b + b^{n}/a)^{1/n}(n=1,2,3, ...).'

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'What kind of shape does point w trace with the following equations:'

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'13\n(3) \\( y^{\\prime}=-\\frac{(5 x+3)^{\\prime}}{(5 x+3)^{2}}=-\\frac{5}{(5 x+3)^{2}} \\)'

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'Divide the circumference into 6 equal parts, mark them clockwise as A, B, C, D, E, F, and place a pebble at point A as the starting point. Roll a die, if an even number appears, move the pebble 2 points clockwise, if an odd number appears, move the pebble 1 point clockwise, continue this game until the pebble returns exactly to point A, which is considered as reaching the goal.'

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'Calculate the total number of permutations when selecting 3 out of 5 numbers to create an even number.'

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'Calculate the sum A+B and the difference A-B of the following polynomials:\n(1) A=7x-5y+17, B=6x+13y-5\n(2) A=7x^3-3x^2-16, B=7x^2+4x-3x^3\n(3) A=3a^2-ab+2b^2, B=-2a^2-ab+7b^2'

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'Convert to standard form. Move the coefficient of \ x^{2} \ which is \ \\frac{1}{3} \ outside the absolute value symbol.'

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'Translate the given text into multiple languages.'

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'Calculate the total number of permutations when selecting 3 numbers out of 5 numbers to create a multiple of 4.'

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'Add the following expressions: (1) 13x + 8y + 12 and x - 18y + 22'

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'(2) Since $\\sin ^{2} \\theta+ \\cos ^{2} \\theta=1$, we have $\\sin ^{2} \\theta=1- \\cos ^{2} \\theta=1-\\left(\\frac{4}{5}\\right)^{2}=\\frac{9}{25}$. $\\theta$ is an acute angle, so $\\sin \\theta>0$, therefore $\\sin \\theta=\\sqrt{\\frac{9}{25}}=\\frac{3}{5}$. Also, $\\tan \\theta=\\frac{\\sin \\theta}{\\cos \\theta}=\\frac{3}{5} \\div \\frac{4}{5}=\\frac{3}{4}$.'

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'Choose 3 different numbers from 0, 1, 2, 3, 4 to form a 3-digit number, denoted as N. Also, let x be the hundreds digit, y be the tens digit, and z be the units digit of this 3-digit number N. Find the following probabilities: (1) The probability that N is a multiple of 3 (2) The probability that y > z'

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'(2) \\[ \\frac{\\sqrt{x} + \\sqrt{y}}{\\sqrt{x} - \\sqrt{y}} = \\frac{(\\sqrt{x} + \\sqrt{y})^{2}}{(\\sqrt{x} - \\sqrt{y})(\\sqrt{x} + \\sqrt{y})} \\] \\[ \egin{array}{l} = \\frac{(\\sqrt{x})^{2} + 2 \\sqrt{x} \\sqrt{y} + (\\sqrt{y})^{2}}{(\\sqrt{x})^{2} - (\\sqrt{y})^{2}} = \\frac{x + 2 \\sqrt{x y} + y}{x - y} = \\frac{(\\sqrt{7} + \\sqrt{5}) + 2 \\sqrt{(\\sqrt{7} + \\sqrt{5})(\\sqrt{7} - \\sqrt{5})} + (\\sqrt{7} - \\sqrt{5})}{(\\sqrt{7} + \\sqrt{5}) - (\\sqrt{7} - \\sqrt{5})} = \\frac{2 \\sqrt{7} + 2 \\sqrt{7 - 5}}{2 \\sqrt{5}} = \\frac{2 \\sqrt{7} + 2 \\sqrt{2}}{2 \\sqrt{5}} = \\frac{\\sqrt{7} + \\sqrt{2}}{\\sqrt{5}} = \\frac{(\\sqrt{7} + \\sqrt{2}) \\sqrt{5}}{(\\sqrt{5})^{2}} = \\frac{\\sqrt{35} + \\sqrt{10}}{5} \\end{array}\\]'

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'Please demonstrate the following rules for addition, subtraction, and multiplication of algebraic expressions.'

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'Please solve a problem related to square root calculation from the exercises in Chapter 1 on numbers and expressions.'

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"Please compute the following expressions by addition and subtraction:\n(1) A+B = (3a^2-ab+2b^2) + (-2a^2-ab+7b^2)\n(2) A-B = (3a^2-ab+2b^2) - (-2a^2-ab+7b^2)\nThis function is referred to as 'CHECK 3'."

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'\ \egin{\overlineray}{c} \\\\ \\mathbb{4} a > 0, b > 0 \\text{ when } \\\\\\ \\\\ \\sqrt{a + b + 2 \\sqrt{a b}} = \\sqrt{a} + \\sqrt{b} \\end{\overlineray} \'

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'State the converse, contrapositive, and inverse of the following propositions.'